Sensitivity of Measurement-Based Purification Processes to Inner Interactions
SSensitivity of Measurement-Based PurificationProcesses to Inner Interactions
Benedetto Militello
Dipartimento di Fisica e Chimica, Universit`a degli Studi di Palermo, Via Archirafi36, I-90123 Palermo, ItalyI.N.F.N. Sezione di Catania
Anna Napoli
Dipartimento di Fisica e Chimica, Universit`a degli Studi di Palermo, Via Archirafi36, I-90123 Palermo, ItalyI.N.F.N. Sezione di Catania
Abstract.
The sensitivity of a repeated measurement-based purification scheme toadditional undesired couplings is analyzed, focusing on the very simple and archetypicalsystem consisting of two two-level systems interacting with a repeatedly measured one.Several regimes are considered and in the strong coupling (i.e., when the couplingconstant of the undesired interaction is very large) the occurrence of a quantum Zenoeffect is proven to dramatically jeopardize the efficiency of the purification process. a r X i v : . [ qu a n t - ph ] A ug ensitivity of Measurement-Based Purification Processes to Inner Interactions
1. Introduction
Preparation of quantum systems is a basic preliminary step in many protocolsand therefore is of fundamental importance in nanotechnology applications, in thefield of quantum information [1], quantum teleportation [2] and even in quantumthermodynamics [3]. If the system is in a mixed state, and we want to prepare it intoa pure state, no unitary evolution is helpful. The simplest way to obtain a pure statefrom a non pure one is to perform a measurement on the system in order to exploit thewave function collapse. Nevertheless, more advanced techniques based on measurementsexist, one of which consists in performing Quantum Non-Demolition measurements(QND) [4, 5, 6, 7, 8]. This scheme is based on the idea that the system that we want toinitialize (we will address it main system) is coupled to an auxiliary system through anHamiltonian that commutes with the free Hamiltonian of the system (in order to avoidback action), and in addition the auxiliary system is repeatedly measured. Alternatively,one can relax the condition of commutation between the free system Hamiltonian and theinteraction between the main and the auxiliary system, following the scheme in Ref. [9].According to such approach, the system to be prepared is coupled to a repeatedlymeasured one, without requiring commutation of the interaction with free Hamiltonians.The net result of this process is the ‘extraction’ of pure states from the initial condition ofthe main system. Starting from this scheme the possibility of distilling entangled statesbetween distant systems, independently from the initial conditions, has been broughtto light [10]. This procedure has been also explored from the theoretical point of viewin several directions: the influence of quantum noise during the extraction process hasbeen analyzed in detail [11, 12, 13], non monotonic behaviors of the purity of the state ofthe system during the extraction process have been brought to light [14]. Moreover, thepossibility of extracting interesting superpositions of angular momentum eigenstatesfor two oscillators through repeated measurements on a two-level system has beentheoretically proven in the field of trapped ions [15]. The possibility of totally controllinga qubit state (its purity, energy, etc) through repeated measurments on an ancilla systemhas also been theoretically demonstrated [16]. Very recently, generation of long-livedsinglet pairs in a nuclear spin ensemble coupled to the electron spins of a NitrogenVacancy center in diamond has been proposed [17]. Over the years, generalizations ofthe repeated-measurement based purification scheme have been proposed: we mentionschemes involving iterative operations (not necessarily measurments) [18] and a two-stepmeasurements scheme [19]. It is the case to underline that very recently many paperson purification protocols, in different physical contexts, have appeared in literaturewitnessing the importance and the actuality of this topic [20, 21, 22, 23].Though the purification scheme of Ref. [9] is based on repeated measurements atregular time intervals, thus recalling the pattern of the quantum Zeno effect (QZE),the time interval between two measurements is typically not that small as requested forQZE [24, 25, 26].In Ref. [27] the possibility of extracting entangled states has been investigated in ensitivity of Measurement-Based Purification Processes to Inner Interactions
2. Framework
Here we summarize the purification scheme introduced in Ref. [9]. Consider a system Sthat we want to initialize and an auxiliary system X that is interacting with the former,repeatedly measured at regular time intervals and always found in the same state, say | ψ (cid:105) X . The effective non unitary dynamics of system S is well described by the followingoperator: V ( τ ) = X (cid:104) ψ | e − i Hτ | ψ (cid:105) X , (1)where H is the total Hamiltonian governing the dynamics of S + X and τ is the timeinterval between two measurements. After N steps, the effective dynamics is given by V ( τ ) N . Let us denote by λ , ..., λ M the eigenvalues of V ( τ ) (ordered in such a waythat | λ i | ≥ | λ i +1 | ), by | λ (cid:105) , ..., | λ M (cid:105) the right eigenvectors and by (cid:104) ˜ λ | , ..., (cid:104) ˜ λ M | the left ensitivity of Measurement-Based Purification Processes to Inner Interactions (cid:104) ˜ λ i | λ j (cid:105) = δ ij , so that V ( τ ) N = (cid:80) k λ Nk | λ k (cid:105)(cid:104) ˜ λ k | . If the system Sis initially in the state ρ , after N steps it will be in the state ρ ( N τ ) = V ( τ ) N ρ [ V † ( τ )] N = (cid:88) kj (cid:104) ˜ λ k | ρ | ˜ λ j (cid:105) ( λ k λ ∗ j ) N | λ k (cid:105)(cid:104) λ j | , (2)which, under the hypothesis | λ | > | λ | and for large enough N , will be approximatedby the following non normalized state: ρ ( N τ ) = (cid:104) ˜ λ | ρ | ˜ λ (cid:105)| λ | N | λ (cid:105)(cid:104) λ | , (3)where the lack of normalization expresses the fact that the procedure is a conditionalone. Therefore we can reformulate this result by saying that the system will be fund inthe state | λ (cid:105) with a probability P ( N, τ ) = (cid:104) ˜ λ | ρ | ˜ λ (cid:105)| λ | N . (4)The success of this procedure depends on three factors: (i) the state | λ (cid:105) must bean interesting state (this depends on the specific needs we have); (ii) the number of stepsrequired to extract | λ (cid:105) must be not too large (this depends on the ratio | λ /λ | : thesmaller this ratio the faster the process); (iii) the probability of success should not go tozero, which is related to the fact that | λ | N must be non vanishing. This last conditionis realized either through the realization of condition (ii) or by fulfilling the condition | λ | ≈
1. In particular, the case | λ | = 1 is said case of optimal extraction and exhibitsstability with respect to the number of steps, being | λ | N = 1, ∀ N . Nevertheless, itis important to note that, if the amount of entanglement in the extracted state saysus that the we can obtain entanglement and a higher stability allows to obtain a nonvanishing success probability for the purification process, it is the efficiency parameterthat determines the very possibility of extracting something. In fact, even in the presenceof high entanglement and optimal stability, a low efficiency means that the process willlast a very long time, and if the second eigenvalues λ has the same modulus of λ then one lose the possibility of extracting anything. In other words, the very possibility(even with low probability) of extracting something (whether an entangled state or not)is given by a non vanishing efficiency.
3. The Model
Here we consider a three interacting-qubit system, one of which is repeatedly measuredin order to purify the state of the other two, and extract entangled states. Moreover,we introduce suitable quantities (witnesses) to quantify the degree of extractedentanglement, the efficiency and the stability of the extraction process.
In Ref.[27] it has been investigated the possibility of extracting entanglement betweentwo systems, say A and B through the interaction with a third system X which is ensitivity of Measurement-Based Purification Processes to Inner Interactions | ψ (cid:105) X .The Hamiltonian of such three-qubit system is the following: H = (cid:88) k =A , B , X ω σ ( k ) z + (cid:88) j =A , B ( (cid:15) σ ( k )+ σ (X) − + h.c. ) . (5)(In Ref.[27] the case of a different free Bohr frequency for the two-level system X isinitially considered, but then the author focus on the homogeneous model with thethree ω ’s all equal.)The two terms of the Hamiltonian will be addressed as H (the free part) and H AXB (the term of interaction between X and the other two subsystems).Now we want to consider the possibility of adding interaction terms between thesubsystems A and B. We consider the following: H AB = η e i φ σ ( A )+ σ (B) − + h.c. , (6)which contains a direct interaction between such subsystems. Following the typical scheme of purification previously mentioned, we assume thatthe system X is repeatedly measured and found in the same state | θ, φ (cid:105) X = cos θ | ↑(cid:105) + e − i φ sin θ | ↓(cid:105) . The predictions about the extraction are given by diagonalizing theoperator V ( τ ) = X (cid:104) θ, φ | U ( τ ) | θ, φ (cid:105) X , (7)which will be our reference model.The eigenstate | λ (cid:105) corresponding to the largest eigenvalue (in modulus) of V ( τ )is the one that will be extracted if the ancilla system X is repeatedly measured every τ and found in the same state. Since we want to extract an entangled state (preferablya maximally entangled state) we need to evaluate this feature. To this scope we canuse the purity of the reduced density operator: E ( ρ AB ) = 2(1 − P (tr A ρ AB )). We thenintroduce the parameter measuring the extracted entanglement as follows:Υ = E ( | λ (cid:105)(cid:104) λ | ) . (8)The ratio between the largest ( λ ) and the second largest ( λ ) eigenvalue tells usthe rapidity of the extraction process: the higher | λ /λ | , the smaller the number ofsteps required to extract the state. Therefore, here we introduce the efficiency as:Λ = 1 − | λ /λ | . (9)Of course, if the parameter Λ is equal to 0, there is no extraction of a single state. Tohave extraction we need this parameter to be nonzero. To have efficient extraction weneed this ratio to be close to unity.We also introduce the stability parameter,Σ = | λ | , (10)which should approach 1 to have an optimal extraction, otherwise the probability ofsuccess of the process will diminish at every step. ensitivity of Measurement-Based Purification Processes to Inner Interactions
4. Sensitivity to Undesired Couplings
In Fig.1 are shown the amount of entanglement of the extracted state (Υ), the efficiency(Λ) and the stability (Σ) of the process of extraction in a case with η = 0, which isessentially the case analyzed in detail in Ref.[27]. It is clear that there are several points(areas) where the entanglement of the extracted state is very high (dark blue regions)and correspondingly the efficiency and the stability are high too. This numerical resultis in perfect agreement with the theoretical analysis developed in Ref.[27], where pointsof optimal extraction of maximally entangled states have been found.In the following subsections, we will consider the effects of the perturbationspreviously described. In particular we will explore the weak and the strong regime. An additional coupling H AB can alter the results of the extraction process, even for smallvalues of η . In Figs. 2,3,4 are shown the discrepancies of the amount of entanglementof the extracted state, of the efficiency and of the stability of the process, as functionsof θ and τ , for particular values of η , φ and (cid:15) , with respect to the η = 0 case. Bluezones indicate improvements (higher entanglement, efficiency or stability, depending onthe case), red zones indicate worsenings, while white areas indicate zero difference.In Fig. 2 it is considered the case η/(cid:15) = 0 .
01 and φ = 0. It is well visible that thereare wide zones white colored, meaning that there are a lot of cases (values of θ and τ ) where the process results insensitive to the presence of the inner coupling. Anyway,differences are present for entanglement, efficiency and stability. In particular, as forthe efficiency, it is well visible that for small values of the variable τ (say for (cid:15)τ < H AB .Fig. 3 describes almost the same situation of Fig. 2 except for the phase, whichnow assumes the value φ = π/
4. This modified value of φ produces visible differences,which are very significant in the efficiency, where the white zones are wider than in the φ = 0 case.In Fig. 4 it is shown only the variation of the extracted entanglement for the φ = π/ η = 0 case. As for the entanglement, in this case thereare only blue zones, suggesting that for small η and φ = π/ It is pretty intuitive that in the strong coupling limit ( η (cid:29) (cid:15) ) the disturbance from H AB is more significant, and the patterns for entanglement, efficiency and stability are verydifferent from this one case for η = 0. In particular, we have observed that for increasingvalues of η the efficiency of the extraction process approaches zero almost everywhere, ensitivity of Measurement-Based Purification Processes to Inner Interactions (a) (b) (c) Figure 1. (Color online). Entanglement Υ (a), Efficiency Λ (b) and Stability Σ (c)of the extraction process as functions of (cid:15)τ and θ/π , for ω/(cid:15) = 2, η = 0. (All plottedquantities lie in the range [0 , Figure 2. (Color online). Entanglement Υ (a), Efficiency Λ (b) and Stability Σ (c) ofthe extraction process as functions of (cid:15)τ and θ/π , for ω/(cid:15) = 2, η/(cid:15) = 0 .
01 and φ = 0.(White color corresponds to a discrepancy smaller than 0 .
01; light blue (red) means anincrease (diminish) between 0 .
01 and 0.1; dark blue (red) means a discrepancy higherthan 0 . i.e., for every values of θ and τ . Figs. 5 and 6 show this phenomenon in a clear way.Moreover, also in this case a sensitivity to the phase φ is very well visible.In Fig. 5 the φ = π/ η : η/(cid:15) = 1, η/(cid:15) = 5and η/(cid:15) = 20, which show that the higher η the smaller the efficiency (whiter picture).We have made plots corresponding to values of η/(cid:15) higher than 20, but they are notreported here, since they are simply white rectangles. Fig. 6 shows the behaviour of theefficiency for the φ = 0 case. The effect of a diminishing efficiency is still present, butthis time we need higher values of η to obtain something comparable to what happensfor φ = π/
5. Theoretical analysis
A complete theoretical explanation of the behaviour of the witness quantities in thedifferent regimes (different values of η and φ ) would require detailed mathematicalanalysis of the V ( τ ) operator. Though in our case it is a 4 × ensitivity of Measurement-Based Purification Processes to Inner Interactions (a) (b) (c) Figure 3. (Color online). Entanglement Υ (a), Efficiency Λ (b) and Stability Σ (c) ofthe extraction process as functions of (cid:15)τ and θ/π , for ω/(cid:15) = 2, η/(cid:15) = 0 .
01 and φ = π/ .
01; light blue (red) means anincrease (diminish) between 0 .
01 and 0.1; dark blue (red) means a discrepancy higherthan 0 . Figure 4. (Color online). Entanglement Υ of the extraction process as functions of (cid:15)τ and θ/π , for ω/(cid:15) = 2, η/(cid:15) = 0 .
01 and φ = π/
2. Graphics for Efficiency and Stabilityare omitted, since they are white rectangles, meaning that there are no significantdiscrepancies with the η = 0 case. (White color corresponds to a discrepancy smallerthan 0 .
01; light blue (red) means an increase (diminish) between 0 .
01 and 0.1; darkblue (red) means a discrepancy higher than 0 . Figure 5. (Color online). Efficiency of extraction Λ as a function of (cid:15)τ and θ/π ,with η/(cid:15) = 1 (a), η/(cid:15) = 5 (b), η/(cid:15) = 20 (c). Here ω/(cid:15) = 2 and φ = π/ | η/(cid:15) | are not reported here, since theysimply are white rectangles. (All plotted quantities lie in the range [0 , ensitivity of Measurement-Based Purification Processes to Inner Interactions (a) (b) (c) Figure 6. (Color online). Efficiency of extraction Λ as a function of (cid:15)τ and θ/π ,with η/(cid:15) = 1 (a), η/(cid:15) = 25 (b), η/(cid:15) = 100 (c). Here ω/(cid:15) = 2 and φ = 0 in all plots.The plot corresponding to higher values of | η/(cid:15) | are not reported here, since theysimply are white rectangles. (All plotted quantities lie in the range [0 , are very long complicated expressions, and the complete diagonalization is not easy, northe results are readable. Nevertheless, it is possible to reach some conclusions throughsome qualitative arguments.First of all, let us consider the structure of the Hamiltonian: H = ω ω η e i φ (cid:15) η e − i φ ω (cid:15) (cid:15) (cid:15) ω ω η e i φ (cid:15)
00 0 0 0 η e − i φ ω (cid:15)
00 0 0 0 (cid:15) (cid:15) ω
00 0 0 0 0 0 0 0 , (11)which is given with respect to the following basis: | ↑↑(cid:105)| ↑(cid:105) X , | ↑↓(cid:105)| ↑(cid:105) X , | ↓↑(cid:105)| ↑(cid:105) X , | ↑↑(cid:105)| ↓(cid:105) X , | ↑↓(cid:105)| ↓(cid:105) X , | ↓↑(cid:105)| ↓(cid:105) X , | ↓↓(cid:105)| ↑(cid:105) X , | ↓↓(cid:105)| ↓(cid:105) X .In the weak coupling limit, the eigenstates of the Hamiltonian can be found througha perturbation treatment in the parameter η/(cid:15) , where the unperturbed Hamiltonian is H + H AXB and the perturbation is H AB . The unperturbed eigenvalues and eigenstatesare: 3 ω , 2 ω + (cid:15) √
2, 2 ω − (cid:15) √
2, 2 ω , ω + (cid:15) √ ω − (cid:15) √ ω , 0, and | ↑↑(cid:105)| ↑(cid:105) X ,2 − / ( | ↑↑(cid:105)| ↓(cid:105) X + | Ψ S (cid:105)| ↑(cid:105) X ), 2 − / ( | ↑↑(cid:105)| ↓(cid:105) X − | Ψ S (cid:105)| ↑(cid:105) X ), | Ψ A (cid:105)| ↑(cid:105) X , 2 − / ( | ↓↓(cid:105)| ↑(cid:105) X + | Ψ S (cid:105)| ↓(cid:105) X ), 2 − / ( | ↓↓(cid:105)| ↑(cid:105) X − | Ψ S (cid:105)| ↓(cid:105) X ), | Ψ A (cid:105)| ↓(cid:105) X , | ↓↓(cid:105)| ↓(cid:105) X , respectively, with | Ψ S (cid:105) = 2 − / ( | ↓↑(cid:105) + | ↑↓(cid:105) ) and | Ψ A (cid:105) = 2 − / ( | ↓↑(cid:105) − | ↑↓(cid:105) ). The first order correctionsto the eigenstates are of the order η/(cid:15) , and the corrections to the eigenvalues are: 0,( η/
2) cos φ , ( η/
2) cos φ , − η cos φ , ( η/
2) cos φ , ( η/
2) cos φ , − η cos φ , 0. This means thatfor φ = π/ V ( τ ) operator closer to the η = 0 counterparts, somehow supportingthe numerical result that for φ = π/ η = 0 case. ensitivity of Measurement-Based Purification Processes to Inner Interactions (cid:15)/η . The eigenvaues and eigenstates of the unperturbed Hamiltonian H + H AB are: 3 ω , 2 ω + η , 2 ω − η , 2 ω , ω + η , ω − η , ω , 0 and | ↑↑(cid:105)| ↑(cid:105) X ,2 − / ( | ↑↓(cid:105) + e − i φ | ↓↑(cid:105) ) | ↑(cid:105) X , 2 − / ( | ↑↓(cid:105) − e − i φ | ↓↑(cid:105) ) | ↑(cid:105) X , | ↑↑(cid:105)| ↓(cid:105) X , | ↓↓(cid:105)| ↑(cid:105) X ,2 − / ( | ↑↓(cid:105) + e − i φ | ↓↑(cid:105) ) | ↓(cid:105) X , 2 − / ( | ↑↓(cid:105) − e − i φ | ↓↑(cid:105) ) | ↓(cid:105) X , | ↓↓(cid:105)| ↓(cid:105) X , respectively. Thecorrections to the eigenstates are of the order (cid:15)/η , becoming more and more negligiblewhen η increases; the corrections to the eigenvalues are all zero. Therefore, on theone hand, it is easy to understand that for very large η the influence of H AXB on thedynamics becomes negligible, meaning that the subsystems AB and X can be consideredas decoupled, then jeopardizing the extraction process. (This occurrence can be seenas a generalized QZE [28, 29], in the sense of a Hilbert space partitioning [30, 31, 32].)On the other hand, in this case there is no easy and direct explanation of the phaseeffect consisting in an acceleration of the efficiency diminishing when φ = π/
2. It can beunderstood in terms of a complete diagonalization of V ( τ ), which, however, is beyondthe scope of the present work.
6. Conclusions
In this paper we have reconsidered the purification scheme introduced in Ref. [9], inparticular analyzing the possibility of taking into account additional interactions to aprefixed scheme. We focused on the special regimes of weak and strong coupling.The additional interaction that we have considered, seemingly, should be helpfulfor the establishment of an entaglement between the subsystems A and B. Neverheless,depending on the situation, it can be helpful or harmful to the extraction process. Thenumerical predictions are partly supported by a theoretical semi-quantitative analysisvalid in the weak and strong coupling limit. In this second case, a dramatic dimishingof the efficiency is predicted, and its connection with a generalized quantum Zeno effect(in the sense of an interaction-induced partitioning of the relevant Hilbert space) isdemonstrated.It is worth recalling that originally the QZE has been introduced as the possibilityof hindering a natural decay (of an atom) through repeated pulsed measurements.Subsequently, the possibility of a dynamical inhibition through strong decays or strongadditional couplings has been proven, leading to the notion of a generalized QZE basedon Hilbert space partitioning. Now, in this paper, we have considered the effects ofan additional interaction that can somehow neutralize the effects of repeated pulsedmeasurements on a system. This fact clearly shows how rich is the panorama ofall possible interplays between interactions and iterated measurements, beyond thestandard formulation of the QZE. [1] D. Bouwmeester, A. Ekert, A. Zeilinger,
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